The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville and the Caputo sense. For a general distribution of time orders we provide the fundamental solution, which is a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative, except for the single order case where the two approaches turn out to be equivalent. We consider in some detail two cases of order distribution: Double-order, and uniformly distributed order. Plots of the corresponding fundamental solutions and their variance are provided for these cases, pointing out the remarkable difference between the two approaches for small and large times.

In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as M‐Wright function, entering as a probability density in a relevant class of self‐similar stochastic processes that we generally refer to as time‐fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time‐integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the M‐Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast types.